For homogeneous saturation, as S varies while porosity remains fixed, the ratio does not change significantly until .At that point, increases dramatically and therefore decreases dramatically. Similarly, as , the only changes in over most of the dynamic range of S are in , which increases linearly with S. Then, when is almost at its maximum value, increases dramatically, causing the ratio to decrease dramatically. Thus, does not change monotonically with S, but first increases a little and then decreases a lot. These two ratios may be conveniently compared by plotting data from various rocks and man-made porous media examples in the (, )-plane [see Figure 1(b) and Figure 2]. We see that, when data are collected at approximately equal intervals in S, the low saturation points will all cluster together with nearly constant and small increases in , but the final steps as lead to major decreases in both ratios. The resulting plots appear as nearly straight lines in this plane, with drained samples plotting to the upper right and fully saturated samples plotting to the lower left in each of the examples shown in Figure 2. The remaining ratio has the simplest behavior, since increases monotonically in S, and does not change. So is a monotonically increasing function of S, and therefore can be considered a useful proxy of the saturation variable S. [Compare Figures 1(c) and 1(d), and see Figure 3.]
Figure 2(a) includes the same sandstone data from Figure 1, along with other sandstone data. Similar data for five limestone samples (Cadoret et al., 1998) are plotted in Figure 2(b). The straight line correlation of the data in the sandstone display is clearly reconfirmed by the limestone data. Numerous other examples of the correlation have been observed. [Fully dry and fully saturated examples are shown here for some of these examples in Figures 2(c) and 2(d), for which partial saturation data were unavailable.] No examples of appropriate data for partially saturated samples have exhibited major deviations from this behavior, although an extensive survey of available data sets has been performed for materials including limestones (Cadoret et al., 1998), sandstones (Murphy, 1984; Knight and Nolen-Hoeksema, 1990), granites (Nur and Simmons, 1969), unconsolidated sands, and some artificial materials such as ceramics and glass beads (Berge et al., 1995). This straight line correlation is a very robust feature of partial saturation data. The mathematical trick that brings about this behavior will now be explained.
Consider the behavior as increases for fixed S. Two of the parameters ( and )decrease as increases, but at different rates, while the third () can have arbitrary variation. [Recall (Bourbié et al., 1987) that rigorous bounds on the parameters are: , ,, and .] To understand the behavior on these plots in Figure 1 as changes, it will prove convenient to consider polar coordinates (r,), defined by
r^2 = w^4()^2 + ()^2, and
= w^2, where w is an arbitrary scale factor with dimensions of velocity (chosen so that r is a dimensionless radial coordinate for plots like those in Figure 1). Now, if in addition we choose w to be sufficiently large so that vs/w << 1 for typical values of vs in our data sets, then, using standard perturbation expansions, we have
r = w^2 (1 + v_s^4w^4)^12 w^2(1 + v_s^42 w^4) and
= ^-1(v_s^2w^2) v_s^2w^2. Thus, the angle is well approximated by the ratio in (thetasimeq), which depends only on the shear velocity vs. We know the shear velocity is a rather weak function of saturation [e.g., Figure 1(a)], but a much stronger function of porosity [see, for example, Berge et al. (1995)]. So we see that the angle in these plots is most strongly correlated with changes in the porosity. In contrast, the radial position r is principally dependent on the ratio , which we have already shown to be a strong function of the saturation S, especially in the region close to full liquid saturation. This analysis shows why the plots in Figures 1(b) and 2 look the way they do and also why we might be inclined to call these quasi-orthogonal (polar) plots of saturation and porosity. Because of the function these plots play in our analysis, we will call them the ``data-sorting'' plots.
In contrast, the plots in Figure 3 contain information about fluid spatial distribution, as will be discussed at greater length later in this paper. The bulk modulus Kf contains the only S dependence in (Gassmann). Thus, for porous materials satisfying Gassmann's homogeneous fluid conditions and for low enough frequencies, the theory predicts that, if we use velocity data in a two-dimensional plot with one axis being the saturation S and the other being the ratio ,then the results will lie along an essentially straight (horizontal) line until the saturation reaches (around 95% or higher), where the curve formed by the data will quickly rise to the value determined by the velocities at full liquid saturation. On such a plot, the drained data appear in the lower left while the fully saturated data appear in the upper right. This behavior is illustrated in Figure 3(a) for Espeil limestone. The behavior of the other plots in Figure 3 will be described below.
Before leaving this discussion of homogeneous saturation, we should note that there is one laboratory saturation technique for which it is known -- from direct observations (Cadoret et al., 1998) using x-ray imaging -- that very homogeneous liquid-gas mixtures will generally be produced. This method is called ``depressurization.'' When such data are available (see Figure 3), we expect they will always behave according to the Gassmann-Domenico predictions. In contrast, the more common approach which produces drainage data is less predictable, since the manner and rate of drainage depend strongly on details of particular samples -- especially on surface energies that control capillarity and on permeability magnitude and distribution. Thus, the drainage technique can produce homogeneous saturation, or patchy saturation, or anything in between.